1

Macroeconomic Dynamics Suvey

Generalizations of optimal growth theory: stochastic models, mathematics, and meta-synthesis

Stephen Spear1 and Warren Young2

1 Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA 15213, [email protected] 2 Dept of Economics Bar Ilan University, 52900 Ramat Gan, Israel, [email protected] 2

Proposed Running Head: Stochastic Growth

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Introduction

In previous papers (Spear and Young 2014, 2015), we surveyed the origins, evolution and dissemination of optimal growth, two sector and turnpike models up to the early 1970s. Regarding subsequent developments in growth theory, a number of prominent observers, such as

Fischer (1988), Stern (1991) and McCallum (1996) maintained that after significant progress in the 1950s and 1960s, economic growth theory

"received relatively little attention for almost two decades" (Fischer,

1988, 329), and that "by the late 1960s early 1970s, research on the theory of growth more or less stopped" (Stern, 1991, 259). Stern went on to say "the latter half of the 1980s saw a rekindling of growth theory, particularly in the work of Romer… and Lucas" (1991, 259), that is to say, in the form of "endogenous growth" models. McCallum, for his part, wrote (1996, 41) "After a long period of quiescence, growth economics has in the last decade (1986-1995) become an extremely active area of research." Moreover, Brock and Mirman’s (1972) paper was the sole

"extension" of Ramsey-Cass-Koopmans to a "stochastic environment" mentioned by McCallum (1996, 49).

4

This paper deals with the evolution of the "classical" growth research program of Ramsey-Cass-Koopmans vintage via its stochastic

"variants" and "generalizations" (Samuelson 1976, note 1). Thus, here we trace the origins and impact of the stochastic generalization that brought about a paradigm shift in modern economics, and still generates significant research in the form of “quantitative macroeconomics”, that is to say, “real business cycle theory” (RBC henceforth), and its metamorphosis into the dynamic, stochastic general equilibrium (DSGE) approaches of both new Classical and new Keynesian vintage. The evolution of endogenous growth approaches and “New” and “Unified” growth models will be dealt with in a separate paper.1

Our focus, then, is on the origins and development of optimal stochastic growth models in continuous-time and discrete-time forms.

The paper is divided into three sections. The first section deals with unpublished and published papers by Phelps (1960a, b; 1961; 1962a, b), and Mirrlees (1965a, b). Phelps' unpublished Cowles Foundation Papers on both continuous-time and discrete-time stochastic optimal growth

(1960b, 1961) are also dealt with in this context—the former never published, the latter the basis for his 1962 Econometrica Paper.

1 It should be noted that the relatively new approach manifest in stochastic endogenous growth models will not be surveyed here, nor will the von Neumann-Gale model in its deterministic and stochastic versions; the evolution and development of these models will be dealt with elsewhere. 5

We then deal with Mirrlees' unpublished papers, dating from 1965, which had significant impact on subsequent work in the area of stochastic optimal growth, such as on the contributions of Merton, Mirman, and

Brock and Mirman respectively. Mirrlees' use of the conceptual and mathematical tools provided by Wiener, Doob, and Ito is also dealt with, as they still influence the financial economics developed by Merton, based upon them. Merton's contributions (1969, 1975) to the continuous- time approach are also dealt with in this section.

The second section deals with the application of the dynamic programming approach of Bellman and Blackwell over the period 1952-

1970, its application to economic planning and growth models, especially by Radner, over the period 1963-1974, and cross-fertilization between

Radner , Brock and Mirman, and Radner's Ph.D student, Jeanjean.

The third section tells the story of how Brock and Mirman developed their watershed approach over the period 1970-1973. It surveys the development of their 1972 JET (1972a) and 1973 IER papers from their origins in their early joint work and Mirman's thesis (1970), through conference presentation, and finally publication. This section also deals with the important, albeit little known third Brock-Mirman paper, that is, their 1971 conference paper published in the volume Techniques of Optimization (1972b). 6

1. Phelps, Mirrlees and Merton: unpublished and published papers,

1960-1975

Phelps: 1960-1962

In his June 1960 RAND paper "Optimal inventory policy for serviceable and reparable stocks", Phelps applied dynamic programming to the problem of determining "a unique stockage policy" regarding serviceable and reparable materials that would correspond to specific

"decision regions". Phelps wrote that "the model developed to treat this sequential decision problem, in being one of the comparatively few two- dimensional dynamic programing models for which the structure of the optimal policy has been ascertained, may be of some methodological interest" (1960a, 4). He went on to apply Bellman's "principle of optimality" to "current" and "future decisions", such that "all future decisions must be optimal" so as to achieve "overall optimality" (1960a,

7). In dealing with "the infinite- stage program", he applied the

"fundamental theorems of dynamic programming for decision processes" outlined by Bellman (1960a, 11).

In December 1960, Cowles Foundation Discussion Paper [CFDP]

101 entitled "Capital risk and household consumption path: a sequential utility analysis" by Phelps appeared. The paper presented what Phelps called "a stochastic process of capital growth" (1960b, 1) in "a continuous 7 time formulation" (1961, 1). Phelps' CFDP 101 was never published. In our view, the paper is important in three respects. First, it was perhaps the earliest paper to apply an ostensibly continuous time approach to optimal stochastic growth, although Phelps' 1960 approach will be shown to be problematic, to say the least. Second, it was also one of the earliest papers to apply a dynamic programming approach and the Bellman

(1957) "principle of optimality" to stochastic optimal growth (1960b, 9).

Third, Phelps’ 1960 CFDP 101 provided the basis for the discrete-time extension of his approach in the form of his subsequent February 1961

CFDP 109, entitled "The accumulation of risky capital: a discrete-time sequential utility analysis". Now, while CFDP 101 (1960b) was cited by

Phelps in his 1961 CFDP (1961, 1-3, 33), only CFDP 109 was mentioned by him in a note in his 1962 Econometrica paper, albeit with its title referred to incorrectly as being "identical" to the Econometrica paper

(1962a, 733 note 6 ). This may explain why CDFP 101 has gone uncited and CFDP 109 sparsely-cited accordingly.

But more is involved here than the fact that Phelps' CFDPs have gone virtually unnoticed until now. In correspondence regarding CFDP

101, Phelps wrote (23 September 2014): "I don't recall any reaction to CF

101 at all… I did not continue with the work that started in CF 101 8 because it was not clear to me that I had the analytical tools to push it any farther".

With regard to the relationship between CFDP 101, CFDP 109 and his Econometrica 1962 paper, he wrote (23 September 2014):

I did not really "switch" to the discrete-time framework. I had

already done all or most of the work for it in my first post-doctoral

job at the RAND Corporation 2. (The great Richard Bellman was

there, as you very likely know, so I finally showed it to him.

"That's trivial", he exclaimed. "The capital stock goes to infinity!"

Of course the whole exercise was aimed at characterizing that path,

solving for the consumption function, etc.) When to my surprise I

ended right back at Yale in September 1960, I worked on the

Golden Rule and what became CF 101. Then, frustrated by how

hard CF 101 was, I prepared the discrete-time paper for what

became the CF 109.

2 Phelps had utilized Bellman's dynamic programming approach in his RAND papers on "Optimal inventory policy for serviceable and replaceable stocks" (1960a), as indicated above, and in his paper "Optimal decision rules for procurement, repair or disposable spare parts" (1962b). 9

Turning first to Phelps' 1960 CDFP 101, a number of points stand out. Phelps described what he was analyzing as a "continuous time formulation" of a "stochastic process" (1960b, 8-10; 1961, 1). He then applied dynamic programming and the optimality principle (Bellman

1957) as the analytical basis for his approach (1960b, 9-17). As Phelps put it (1960b, 9, 16-17): "In what follows we take our inspiration from

Chapter 9 of Bellman… our continuous-time process can be viewed as the limiting case of a discrete-time process in which the length of each period goes to zero while the number of discrete periods goes to infinity…"

A close reading of Phelps' 1960 CFDP 101 reveals the difficulties

Phelps faced in attempting to apply and develop his ostensibly

"continuous" approach. First of all, he stated that in his model "capital gains and losses occur in unit amounts… fluctuations in the capital stock occur at random times… Thus capital grows according to a discontinuous Markov process" (1960b, 2) [our emphasis]. In other words this meant that the expected time path of wealth shocks was continuous, even though the underlying random shocks were discontinuous, that is to say, a "mixed model" rather than a pure continuous-time approach.

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Second, the result Phelps outlined in his 1960 CFDP 101 was that a consumer could end up "ruined" (1960b, 3) due to "absorbing" states and

"cyclic" states, although he noted that there could be both low and high capital persistence states that occur with positive probability, given the finite time horizon he assumed (1960b, 21-22). If, however, Phelps had pushed this through for an infinite horizon setting, he would have ended up with an ergodic distribution of the recurrent states, and the high or low capital "traps" (1960b, 21) would not occur, because there would be no absorbing states. Now, while Phelps does in fact consider what happens when the time horizon goes to infinity, and therefore the "ruin" probability goes to zero, he does not discuss this in terms of ergodic theory.

We think that the reason for Phelps abandoning the ostensibly continuous-time setting of his 1960 CFDP 101 for the discrete-time framework of his 1961 CFDP 109, may have been to get away from the awkwardness of mixing a discrete stochastic process with a continuous time model. We surmise that he could have chosen to work with a continuous Markov process, as the works of Wiener (1923), Doob (1953), and Ito (1951) respectively—i.e. stochastic calculus—were available at the time, but the move to a continuous stochastic process would, in turn, rule out purely transient and cyclic states, and thereby "trap" behavior, 11 even with finite time horizons, as the continuity of the random shock process would make things look as though there were infinitely many instances of time, so that ergodic behavior would come into play, something Phelps may not have wanted in the context of what he was trying to do in his 1960 CFDP 101.

As noted, Phelps CFDP 109 entitled "The accumulation of risky capital: a discrete-time sequential utility analysis" appeared in February

1961. In the opening paragraph, Phelps wrote (1961, 1): "A continuous- time formulation of the same problem was presented in a previous

Cowles Foundation Discussion paper". His October 1962 Econometrica paper, based on CFDP 109, was entitled "The accumulation of risky capital: a sequential utility analysis". And, although not cited in the references, Phelps did mention CFDP 109 as an "earlier version of this paper" (733, note 8). Indeed, there were many additions, changes and elisions made in the text of CFDP 109 as published in Econometrica.

But perhaps the most important addition seen in the 1962 version is the statement made by Phelps, which set out the "vehicle of analysis" for all further work done on discrete optimal stochastic growth models. As he put it in his opening paragraph (1962a, 729): "The vehicle of analysis is a stochastic, discrete-time dynamic programming model that postulates an expected lifetime utility function to be maximized."

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Mirrlees: 1964-1974

According to Fischer and Merton, “Following the early unpublished work by Mirrlees (1965), Brock and Mirman (1972; 1973), Bourguignon

(1974), and Merton (1975) among others, extended the neoclassical growth model to include uncertainty about technological progress and demographics” (1984, 58). Over a decade earlier, in their seminal paper,

Brock and Mirman wrote (1972a, 481)

The only work which is known to the authors which attempts to

generalize deterministic optimal growth models to uncertainty… is

the work of Mirrlees [1965]. Works by Phelps [1962] and Levhari

and Srinivasan [1969] are somewhat related to optimal growth

under uncertainty…

They went on to also cite Merton (1969) as using “techniques similar to those employed by Mirrlees” (1972a, 482). We will deal with the Brock-

Mirman interpretation of these authors below. At this point, we note that

Olson and Roy, in their survey of stochastic optimal growth, wrote (2006,

298 note 1): “there is a large literature on stochastic growth in continuous time that built on Merton’s…early work [RES 1975]”, and also referred to Brock and Macgill (1979). They did not, however, refer 13 to the seminal, albeit unpublished paper by Mirrlees entitled "Optimum accumulation under uncertainty" (1965a, b), which was cited by many who dealt with the stochastic optimal growth model, as will be seen below.

In his contribution to the IEA conference volume entitled Allocation

Under Uncertainty, edited by Dreze, and published in 1974, Mirrlees wrote (1974,36) "In the theory of optimum growth it has been found that models with discrete time are easier to treat rigorously than models with continuous time. But continuous-time models often have the advantage of providing simpler results ".

Mirrlees worked with both non-stochastic and stochastic models.

After his Cambridge Ph.D dissertation entitled "Optimum planning for a dynamic economy" (Mirrlees, 1964a) he attended a number of

Econometric Society and other meetings. At the September Zurich 1964

Econometric Society meeting, he presented a paper entitled "The structure of optimum policies in a macro-economic model with technical change"(1964c), which later became the core of his 1967 RES paper

"Optimum growth when technology is changing "(1967, 96, 124); in essence a continuous non-stochastic extension of Cass's approach, with exogenous technical change. 14

Two months earlier, in July 1964, Mirrlees attended the Rochester conference “Mathematical models of economic growth”, sponsored by the SSRC, and led by McKenzie. At this conference, according to

McKenzie’s account (1998, 5), Mirrlees presented two papers; a two- sector extension of Uzawa (1964), and an attempt “to extend the Ramsey model to the case of uncertainty”. McKenzie wrote that in this paper,

Mirrlees “apparently met a snag he was unable to overcome”, and that while McKenzie later tried to discover “what the difficulty was”, Mirrlees

“could not recall, and had lost the paper” (1998, 5). When asked about his

Rochester presentation, Mirrlees replied (4 Sept 2014): “I recollect that at the Rochester Conference I was trying to deal with the discrete-time model, also with the assumption of a random factor multiplying output, not stationary as in the later paper published in the Dreze book . I didn’t succeed after the conference any better than I had in Rochester.” In a subsequent communication, (30 December 2014), Mirrlees also recalled the title of his Rochester paper to be "Optimum economic policies under uncertainty"(1964b), and that in “this unfinished paper” he “was trying to work out optimal saving under uncertainty for a discrete-time infinite- horizon model, and finding it remarkably difficult to get any results”

Mirrlees also attended the July 1965 Stanford MSSB Conference on

“Optimal Economic Growth” conducted by Arrow, and initiated by 15

McKenzie himself “as the economics member of the MSSB”. According to McKenzie’s recollections, Mirrlees presented a paper on “planning in mixed economies with ‘surplus labor’”, and also led a roundtable discussion “on population and growth” (1998, 6-7).

Mirrlees’ paper entitled “Optimum accumulation under uncertainty”, was presented at the First World Congress of the Econometric Society held in Rome, in September 1965 (1965a), revised and circulated in

December 1965 (1965b). And, as noted, this paper, albeit unpublished, was cited by leading growth theorists, and many of those who developed stochastic optimal growth.

A comparison of the abstract of the September 1965 version of

Mirrlees’ paper (Econometrica, Supplementary Issue, 1966), with parallel text and equations in the December 1965 version shows only very minor differences.

Two interesting points emerge from Mirrlees’ December 1965 paper. The first is its use of Bellman’s dynamic programming in a stochastic setting (1965b, 27; Bellman, 1957). The second is that it also contains a section entitled “Orders of magnitude”, which comments on the “quantitative implications” of the theory he presents in the paper, linking it to “the interesting values of the parameters and the capital- output ratio” he dealt with in his 1964 Zurich paper (1965b, 44). 16

Mirrlees’ unpublished December 1965 paper was cited by most optimal growth theorists and others (McKenzie, 1968; Levhari and Srinivasan,

1969; Stiglitz, 1969; Merton, 1969; Sandmo, 1970; Dobell, 1970; Brock and Mirman, 1972a, 1972b; Merton, 1973; Leland, 1974; Merton, 1975;

Samuelson, 1976; Fischer and Merton, 1984). What is strange is that two references said that the paper was “forthcoming” in Econometrica

(Sandmo, 1970; Dobell, 1970).

When asked about this, Mirrlees replied (28 June 2014):

Econometrica more or less (I don't remember precisely) accepted

the paper subject to revision. When I came to revise it, I found an

error in the existence proof, and didn't succeed in correcting it.

Later I convinced myself that the existence claim was false, and it

is old unfinished business for me to sort out the fundamental

existence problem with the model used in the paper. The worrying

point is that utility discounting seemed to me not enough to deal

with it. By the time I realized the problem, my interests had moved

well away from optimum accumulation, but I am ashamed of not

having sorted the problem out. I have never seen a paper that does.

17

Mirrlees also attended the July 1971 IEA Bergen conference, where he gave a paper entitled "optimal growth with uncertainty" (Merton,

1975), which he revised and expanded in May 1972, changing the title to

"optimum growth and uncertainty" (Mirrlees 1974, 36), and eventually publishing it in the 1974 IEA conference volume edited by Dreze,

Allocation under Uncertainty, with the title “Optimum accumulation under uncertainty: the case of stationary returns to investment”.

Merton 1969-1975

Merton submitted his MIT Ph.D thesis, supervised by Samuelson, and entitled "Analytical optimal control theory as applied to stochastic and non- stochastic economics", in September 1970. The thesis consisted of five chapters, three of which were published in 1969. Chapter II, entitled “Lifetime portfolio selection under uncertainty: the continuous- time case” was published in August 1969 in the Review of Economics and

Statistics. In the introduction, Merton wrote “Phelps (1962) has a model used to determine the optimal consumption rule for a multi-period example where income is partly generated by an asset with an uncertain return. Mirrlees (1965) has developed a continuous time optimal consumption model of the neoclassical type with technical progress a random variable” (1969, 247). In the concluding section, Merton wrote

(1969, 256-257): 18

A more general production function of the neoclassical type could

be introduced to replace the simple linear one of this model.

Mirrlees (1965) has examined this case in the context of the growth

model with…technical progress a random variable. His equations

(19) and (20) correspond to my equations (35) and (37) with the

obvious proper substitution for variables.

Thus, the technique employed for this model can be extended to a

wide class of economic models. However, because the optimality

equations involve a partial differential equation, computational

solution of even a slightly generalized model may be quite

difficult.

Two things should be pointed out here. First, Merton was referring to equations in the unpublished December 1965 paper by Mirrlees. It would seem, then, that a number of leading economists, and especially growth theorists, were familiar with this paper. Indeed, in the copy of the

Mirrlees’ paper provided to the authors by Merton, there are marginal notes made by Samuelson, who seems to have given a copy of the paper to Merton. Secondly, regarding his “equation 19”, Mirrlees wrote (1965,

13):

It is also interesting to note that (19) generalizes what is often

called the Keynes–Ramsey formula for optimum policy… In that 19

case, of course, (19) solves the problem, provided the side

conditions are satisfied. Mathematically (19) and (20) are a fairly

decent pair of partial differential equations. The trouble is that the

side conditions are of such an odd kind, and that creates serious

computational difficulties”.

In the 1971 published version in Journal of Economic Theory of

Chapter 5 of his thesis--which had earlier been presented at the 2nd World

Conference of the Econometric Society—entitled “Optimum consumption and portfolio rules in a continuous time model”, Merton wrote (1971, 412): “By the introduction of Ito’s Lemma and the

Fundamental Theorem of Stochastic Dynamic Programming …we have shown how to construct systematically and analyze optimal continuous- time dynamic models under uncertainty.”

Merton’s seminal paper on growth “An asymptotic theory of growth under uncertainty” was published in the Review of Economic Studies in

July 1975. It started as a paper presented “in various forms” at venues such as the December 1971 Yale NBER growth conference, the March

1973 Rochester mathematical economics seminar, and the April 1973 mathematical economics seminar at Columbia (Merton 1975, 375). It was subsequently circulated as MIT Sloan School working paper 673-73 in August 1973 (Merton, 1973). The paper was submitted to the Review 20 of Economic Studies a month later, and finally accepted in May 1974

(Merton, 1975).

A comparison between the 1973 working paper and 1975 published versions shows a number of significant additions in the published paper in the form of explanatory notes; a result both of editorial suggestion

(1973, 11; 1975, 383 note 1), and Merton’s efforts to explain his use of various methods, and issues in the paper. For example, he added notes explaining his use of the Bellman approach and optimality in solving dynamic programming problems in continuous time, boundary conditions, and generalizations of maximization to uncertainty (1973,

14,16; 1975, 384 notes 1 and 2, 386, note 1).

In the introduction to his paper (1973, 1; 1975, 375), Merton cited works that dealt with “capital accumulation under uncertainty” and “the optimal consumption-savings decision under uncertainty”, based upon “a given linear production technology” [Phelps (1962), Levhari and

Srinivasan (1969), among others]. He also cited the unpublished

December 1965, and the early version of the 1964 conference volume paper, by Mirrlees, as examples of dealing with “the stochastic Ramsey problem and a continuous time neoclassical one-sector model subject to uncertainty about technical progress”. He went on to describe the work of Brock and Mirman (1972), and Mirman (1973) as “important 21 contributions”, albeit having “little to say about the specific structure” of

“steady state” or “asymptotic distributions” regarding the “capital-labor ratio”, when “outcomes are uncertain”. And this, as against the model he proposed “where the dynamics of the capital-labor ratio” is “described by a diffusion-type stochastic process”.

The Brock-Mirman assessment of Mirrlees-Merton will be dealt with below, in the section on the development of the work of Brock and

Mirman from 1970 onwards. At this point, suffice it to say that Merton

(1969, 248; 1971, 412; 1973, 6; 1975, 377), as Mirrlees before him (1965,

3) had utilized an approach based upon “a generalized theory of stochastic differential equations developed by Ito”, and extended by Ito and McKean “which is applicable to diffusion processes”.

Ito calculus is a variety of stochastic calculus, which extends the normal operations of calculus – differentiation and integration – to stochastic processes. Unlike smooth (i.e. continuous and continuously differentiable) functions, stochastic processes can be discontinuous and non-differentiable, manifesting sudden random jumps in value.

At its core, stochastic calculus operations follow conventional operations, with the difference being the recognition that over even small intervals of time, non-vanishing changes in the value of the stochastic process can occur, and hence need to be included in computations of 22 things like rates of change, in the case of differentiation, or weighted suitably in taking the limits of partitions that define the Riemann integral.

The specific version of the stochastic calculus applied by Mirrlees and Merton, Ito’s calculus, was formulated between 1938 and 1945 by the Japanese mathematician Kiyoshi Ito while he was working at the

National Statistical Office [Ito, 1942, 1951; Ito and McKean, 1964; Ito,

1965].

2. Dynamic programming, economic planning and growth: 1948-73

Bellman, Karlin and Blackwell

As mentioned above, in their survey on stochastic optimal growth,

Olson and Roy (2006) did not cite Mirrlees' seminal, albeit unpublished, paper (1965b). They also did not cite Bellman's famous book Dynamic

Programming (1957), although they did cite Blackwell's paper

"Discounted dynamic programming" (1965). What is important to recall here is that Bellman's 1957 book was based upon his work at RAND and elsewhere, and published and unpublished papers, from 1948 onwards, and this according to Bellman's own recollections and accounts (Bellman,

1984; Bellman and Lee, 1984, 24).

Bellman first visited RAND to attend its 1948 summer program, which was also visited by Morgenstern. Among the other attendees were

Dantzig, Karlin, Tukey, Blackwell, Arrow and Shubick (Assad, 2011, 23

422). Bellman wrote that his introduction to Dantzig's linear programming algorithm while there was his "first exposure to effective numerical solutions", which, as he recalled, "subsequently became a central theme" of his research program (Bellman 1984, 135). He then went to Stanford to take up a position as Associate Professor in its

Mathematics Department. In the summer of 1949, Bellman returned to visit RAND again. This time, at the suggestion of a colleague at RAND, he shifted the focus of his research to "multistage decision processes"

(Bellman 1984, 157; Assad, 2011, 424). During this period, according to his recollections, he had to essentially "find a name for multistage decision processes" (Bellman 1984, 159).

In a July 1951 RAND paper entitled "On a general class of problems involving sequential analysis", Bellman's set out the basis for what he subsequently called "dynamic programming". As he put it (1951, 1):

We wish to discuss a general class of multi-stage problems

involving a sequence of operations …This class of problems is

characterized by the fact that at each time the problem may be

described by a set of parameters which change from operation to

operation, which is to say that each operation performs a mapping

of the parameter space upon itself, and secondly, that the purpose

of the operations is to optimize according to a criterion which has

the important property that after any initial number of operations, 24

starting from the state one finds oneself in, one optimizes

according to the same criterion. …this last point … allows a

mathematical formulation by means of recurrence relations which

are very useful both theoretically and computationally.

Over the period 1952-1957, Bellman produced a significant number of papers on dynamic programming and on its applications to many areas, including to problems in mathematical economics, as will be seen below.

The central message of these papers, as manifest in what he called "the principle of optimality" and the method he developed, as reflected in what became known as the "Bellman equation", eventually appeared in his 1957 monograph. Indeed, as one reviewer said, Bellman's 1957 book brought "under one cover the introduction and development of the theory of dynamic programing, which to a great extent has appeared previously in many papers scattered throughout many journals and pamphlets"

(Newhouse 1958, 788).

Now, one of the problems emanating from the fact that Bellman's

1957 book was, in essence, a compilation of his previously published papers and RAND reports, is that the focus has been on his 1957 book rather than on his earlier work. This, in turn, has led to some misunderstanding regarding the origin of his central message, that is to say, the principle of optimality. For example, Puterman (1994,155) noted 25 that his book "presented the optimality equations and the principle of optimality together with references to his earlier papers (dating back to

1952) which introduced and illustrated many of the key ideas of dynamic programming".

Acemoglu, for his part wrote (2008, 222) "the basic ideas of dynamic programming, including the principle of optimality, were introduced by Richard Bellman in his famous monograph (Bellman,

1957)". And Acemoglu, as Puterman , noted that Karlin (1955) had provided "a formal mathematical structure for the analysis of dynamic programing models" (Puterman 1994, 155), and " a simple formal proof to the principle of optimality" (Acemoglu 2008, 222).

Both Puterman and Acemoglu, however, overlooked the fact that

Bellman's earliest published contribution specifically regarding dynamic programming entitled " On the theory of dynamic programming" appeared in the 1952 Proceedings of the National Academy of Sciences,

"communicated by Von Neumann in June 1952" (Bellman 1952. 716), in which he acknowledges that dynamic programming was "intimately related to the theory of sequential analysis due to Wald [1950] (1952,

717)".3 Had they also looked carefully at Karlin's 1955 paper they would have seen that Karlin (1955, 285) pointed to the fact that the principle of

3 There are both priority and multiple discovery issues regarding Bellman's approach. These were raised by colleagues at RAND, in the early 1950s, and recently, by historians of mathematics. On these issues, see Pesch (2012) and Pesch and Plail (2009, 2012). 26 optimality appeared in Bellman's Econometrica paper, "Some problems in the theory of dynamic programming", published in January 1954

(1954a, 47). Moreover, in July 1954, in his RAND paper P-550, entitled

"The theory of dynamic programming", Bellman again presented the principle of optimality (1954b, 4), as in the published version of this

RAND paper, that appeared in the November 1954 issue of the Bulletin of the American Mathematical Society (1954 c, 504).

Another interesting aspect of the Bellman-Karlin nexus is that prior to Karlin's paper, published in the December 1955 issue of the Naval

Research Logistics Quarterly, Bellman had discussed Karlin's paper

"Some aspects of dynamic programming", presented on 1 September

1955, at the Ann Arbor meeting of the Econometric Society, at a session also attended by Koopmans (Report of 1955 meeting, Econometrica

1956, 208).

What is also important to mention here is that in a number of papers that appeared both before and after his 1957 book, Bellman specifically dealt with the application of dynamic programming to mathematical economics. In 1956, his RAND paper "Dynamic programming and its application to variational problems in mathematical economics ", appeared (1956 a). It was subsequently presented at the Eighth American

Mathematical Society's Symposium in Applied Mathematics, held at the

University of Chicago in April 1956 (1956 b), and was eventually 27 published in the symposium Proceedings in 1958 (1958 [1956 b]). He wrote (1958, 115 [1956 b]):

The purpose of this paper is to discuss some variational problems

arising from mathematical economics, and some of the methods

that can be used to treat these questions both analytically and

computationally.

Since the range of mathematical economics is so extensive—and

indeed the subject possesses no precise boundaries—and since the

array of mathematical techniques which have been borrowed,

begged, stolen, or improvised to cope with this field is so

imposing, we cannot hope to present any adequate survey in any

reasonably sized article. In consequence, we have restricted our

attention to two important and interesting classes of processes,

allocation and smoothing processes, and to a discussion of the

application of the theory of dynamic programming to these

processes.

Later, in March 1963, Bellman's RAND paper," Dynamic programming and mathematical economics" (1963, RM-3539-PR) appeared. Here, he toned down his language a bit, and wrote (1963,

Preface, iii): 28

In this memorandum, the author describes the uses and

contributions of the mathematical theory known as dynamic

programming to certain problems in economics. Examples of these

are the optimal allocation of resources, and multistage decision

processes that involve planning and learning in the face of

uncertainty (i.e., adaptive control processes).

He continued on to say (1963, Introduction, 1)

The functions of a mathematical theory in the scientific field are to

furnish the systematic means of formulation of classes of problems,

to indicate various techniques for their analysis, and to provide

methods for obtaining numerical answers to numerical questions.

At one point in the nineteenth century, serious doubt was expressed

that problems in the field of economics could ever be handled

mathematically. The introduction of the digital computer changed

the situation drastically. Nevertheless, much remains to be done,

and many new approaches must be devised, before we can consider

ourselves to have a firm hold in the domain of mathematical

economics.

29

In this memorandum we outline briefly some of the principal

contributions of the theory of dynamic programming the

formulation, analysis, and computational treatment of economic

processes.

In a series of papers from 1961 onwards, Blackwell also focused on dynamic programming (1961; 1962; 1964 a, b; 1965). Interestingly enough, as early as 1952, Blackwell had assisted Bellman in obtaining a solution to a least one of the fundamental theorems of Bellman's dynamic programming approach (Bellman 1952, Theorem 7, 719). Later, in his

1961 paper "On the functional equation of dynamic programming",

Blackwell demarcated stable, optimal, and stable optimal policy (1961,

274), and extended Bellman's1957 treatment to the case of policy- switching and its effects on optimal policy and stability (1961, 274). In a subsequent paper entitled "Discrete dynamic programming" (1962),

Blackwell turned to simplify the results obtained by Howard (1960) regarding the introduction of discount factors less than and equal to unity into "the general dynamic programming problem" set out in the works, as he put it, of "Dvoretzky, Kiefer and Wolfowitz(1957) , Karlin (1955) and

Bellman (1957)".

In his 1965 paper "Discounted dynamic programming", Blackwell wrote (1965, 2 to 6): 30

Soon after the appearance of Wald's work on sequential analysis,

Richard Bellman recognized the broad applicability of the methods

of sequential analysis, named this body of methods dynamic

programming, and applied the methods to many problems… The

first development of a general theory underlying these methods is

due to Karlin(1955), and a rather complete analysis of the finite

case was given by Howard (1960)… Our formulation of the

dynamic programming problem is somewhat narrower than

Bellman's.

Bellman and Blackwell

We now try to explicate the contributions of Bellman and Blackwell in the context of a discrete time model, but the extensions to continuous time are relatively straight-forward. Let us consider a multi-sector capital model with a single infinite-lived representative agent. The objective function for the planning problem is to maximize an infinite discounted sum of utilities of consumption in each period subject to the constraints that output is produced using labor (fixed) and capital (variable) inputs and that output allocations of consumption goods must be allocated feasibly. Further, assume that the period utility functions are strictly concave, and production functions exhibit constant returns to scale

(CRS). 31

Now, look at a sub-problem where we want to do the maximization over some finite number T periods. In this case, since the objective function is a sum of concave functions, and constraint sets are convex, the standard results from concave programming theory will be applicable, once we recognize that we need a terminal condition on what the period

T+1 vector of capital stocks should be. One approach, which Lucas and

Stokey use to get the standard transversality condition, is to set the terminal stock to zero. If we are interested, however, in looking instead at short-run efficient allocations, we should maximize the sum of utilities up to T plus K(T+1). In between, we can specify whatever terminal condition we like. Once we do this, concave programming theory says that the first-order conditions (more generally, the Kuhn-Tucker conditions) will be necessary and sufficient to characterize the solution.

To get an answer to our original question, we need to let T go to infinity. Doing this formally in a way that answers the original question then requires the techniques of Pontryagin for dealing with the terminal condition and getting the right limiting transversality condition. This is what Cass did in chapter one of his thesis (1965). What would not be correct, though, would be to claim that in the T=infinity case, we have an infinite discounted sum of concave utilities, so that we are in the world of concave programming and the first-order conditions (i.e. the Euler 32 equations) are necessary and sufficient. Indeed, they are not, as

Malinvaud's counter-example to Koopmans showed (1964), even though they are for any finite T. Something is very different in the infinite dimensional case.

To see what is different, one needs to go back to how the concave programming (Kuhn-Tucker theory) result is proven, and what we find when is that it relies, not surprisingly, on applying the Separating

Hyperplane Theorem (i.e. Minkowski's theorem). Now, Minkowski's theorem holds in very general spaces, but it does have one stringent requirement: one of the two disjoint, convex sets being separated must have a non-empty interior. One of things that John Tukey (1942) is famous for is having provided a counter-example to the claim that you can dispense with the non-emptiness requirement in Minkowski’s theorem. He constructed two disjoint convex sets in l2 (classic Hilbert space), neither of which had an interior, and showed that there was no linear functional that would separate them.

Since the discounted sum of utilities specification of the capital model necessarily involves looking at something in the positive orthant of an infinite-dimensional space, the interiority criterion becomes relevant, and a direct application of the Minkowski theorem isn't available. So, we are left with the situation in which the Euler equations are necessary, but 33 not sufficient. What Pontryagin et. al. demonstrate is how to augment this with the right specification of the terminal conditions to pick out the one correct trajectory satisfying the Euler equations, so that the two conditions together are necessary and sufficient.

Now, the forward dynamic programming (DP) algorithm gets around the problem of terminal conditions by embedding the optimization problem in a recursive time-structure where the future looks like a steady- state extension of the past. Specifically, given a function specifying the continuation value of the DP given the values of the state variables one period hence, we are left with a simple, finite-dimensional programming problem. If all of the functions involved are concave, we have a simple, finite-dimensional concave programming problem, and, of course, the

FOCs here are necessary and sufficient.

The genius of Bellman and Blackwell was to recognize that they weren't actually getting rid of the infinities that cause problems in the control theory setting; rather, they were converting the problem of the infinite time-horizon into that of finding one among an infinity of possible continuation functions. In the problems that involve discounting

(or other set-ups that satisfy the Blackwell conditions for a contraction), the functional equation that replaces the infinite discounted sum of utilities turns out to be a contraction mapping, and the Banach theorem 34 therefore applies, yielding a fixed-point to the Bellman equation, and, via the obvious recursion, a solution to the original optimization problem. Again, because the Bellman equation evaluated at the fixed point value function is a finite-dimensional concave programming problem, the FOCs are necessary and sufficient.

The solution that comes out of the DP approach can also be used to demonstrate the sufficiency of the transversality condition in the original problem. The ascendance of the DP approach in growth theory is clearly due to the availability of the contraction mapping results, and their computational tractability. We think this also explains why the profession brushed aside the whole question of the moral implications of discounting so widely discussed in the Vatican Conference volume

(1964), at least until the issue was resurrected in the contemporary discussion of global warming.

Cross-fertilization: Radner, Brock-Mirman, and Jeanjean

As noted above, Phelps was perhaps the first to introduce Bellman's dynamic programming approach in economics in both continuous- and discrete-time settings in his work between 1960 and 1962. Over the period 1963-1974, Roy Radner also advanced the application of dynamic programming in economics. For example, his ONR supported 1963

Technical Report entitled "Notes on the Theory of Planning" utilized dynamic programming which, as he wrote (1963, 2) "is relatively new to 35 the theory of economic planning". He gave accounts of "the dynamic programming valuation function for various functions and programs"

(1963, Lecture 4, sections 2-3). In February 1964, Radner's Berkeley

Center for Research in Management Science Technical Report Number

17 (also supported by the ONR), entitled "Dynamic programming of economic growth" appeared. The abstract of the paper read as follows:

A class of problems of optimal economic growth is formulated in

terms of the functional equation approach of dynamic

programming (Bellman, 1957). A study is made of the continuity

and concavity properties of the state valuation function, i.e., the

function indicating a maximum total discounted welfare (utility)

that can be achieved starting from a given initial state of the

economy. Under suitable conditions this function is characterized

by a certain functional equation. Both the cases of a finite and an

infinite planning horizon are treated, the latter case being discussed

under the assumption of constant technology and tastes. Here

iteration of a certain transformation associated with the functional

equation is shown to provide convergence to the state valuation

function. Exact solutions are given for the case of linear-

logarithmic production and welfare functions.

36

Radner's 1964 Technical Report 17 went virtually unnoticed. It was only cited in his 1966 International Economic Review paper "Optimal growth in a linear-logarithmic economy". An abridged version of the report was published in 1967 under the same title, that is, "Dynamic programming for economic growth" (1967, 111-141). Now, the introductory paragraph in the 1967 abridged version was identical to the abstract of his 1964 Technical Report. However, in contrast to the

"relatively new" description of dynamic programming he used in his 1963

Technical Report 9, in his 1967 paper Radner wrote (1967,111):

The techniques used are familiar to workers in the field of dynamic

programming, although the particular assumptions appropriate to a

study of optimal economic growth differ from those commonly

encountered in other applications (e.g. inventory and replacement

theory)[such as in Phelps, 1960].

Radner continued:

The interest in such an approach must ultimately derive from its

power, if any, in producing new characteristics of optimal growth

and/ or new and more efficient computational techniques.

Over the period 1971-1973, Radner both presented, at various meetings, and published two important papers relating "optimal steady- 37 state" programs and stochastic production. One was presented at the

Mathematical Social Science Board (MSSB) workshop at the University of California, Berkeley, July-August 1971, and published in the Journal of Economic Theory in February 1973. The other was presented at the symposium on mathematical analysis of economic systems at the Fall

1971 meeting of the Society for Industrial and Applied Mathematics

(SIAM), held at the University of Wisconsin-Madison, 11-13 October

1971, and published in Mathematical Topics in Economic Theory and

Computation (1972). In both his 1972 and 1973 papers, Radner used the identical "bibliographic note" in which he cited two papers by Brock and

Mirman, one a paper they presented at the summer 1971 Berkeley MSSB workshop, the other, their now famous 1972 Journal of Economic Theory paper. Radner wrote (1972, 89; 1973, 110-111):

W.A. Brock and L.J. Mirman (1971), (1972) have studied optimal

growth under uncertainty in a model with one commodity and a

sequence of independent and identically distributed states of the

environment. In particular, in the second paper they considered the

problem of optimal stationary programs.

We will return to the MSSB meeting below.

Another work cited by Radner in his SIAM (1972) and JET (1973) papers, was the unpublished September 1972 Berkeley Ph.D thesis of 38

Jeanjean, supervised by Radner, entitled Optimal Growth with Stochastic

Technology in a Multi-Sector Economy. In May 1971 Jeanjean had circulated his Berkeley Center for Research in Management Science

Working Paper 332 entitled "Optimal growth with stochastic technology in a closed economy". Interestingly enough, in 1974, Jeanjean published two additional papers, one in French, which was, in effect, the translation of his thesis (1974a). The other was a paper published in JET entitled

"Optimal development programs under uncertainty: the undiscounted case" (1974b). In the French translation of his thesis (1974a, 98),

Jeanjean cited an unpublished paper by Mirman entitled "The steady state behavior of a class of one sector growth models with uncertain technology" (1970). This paper was later published by Mirman in the

June 1973 issue of JET under the same title. According to Mirman, (9

November 2014, personal communication to authors), he gave the 1970 paper, and other papers, to Radner at the MSSB meeting. This cross- fertilization continued, with Jeanjean's thesis being cited by Brock and

Mirman in their paper in the volume Techniques of Optimization (1972b), and in their 1973 International Economic Review paper " Optimal economic growth and uncertainty: the no discounting case". They reported that Jeanjean had "extended some" of their "results to multisector models" (1972b, 418) and that "Recently our results have 39 been generalized to the multisector case by Jeanjean" (1973, 572), citing

his unpublished thesis accordingly (1972b, 418; 1973, 573).

The summer 1971 Berkeley MSSB workshop on "Theory of markets and uncertainty" was conducted by Radner. The participants were:

William Brock, Leonard Mirman, Peter Diamond, Jerry Green, Theodore

Groves, Werner Hildebrand, Michael Rothschild, Jose Scheinkman,

Michael Spence, Bernt Stigum, and Shmuel Zamir (Cutler, 1973). Now, according to McKenzie (1999, 10), two papers by Brock and Mirman on growth with uncertainty, dealing with the undiscounted and discounted cases, were given at Radner's workshop. There are however, some problems regarding McKenzie's account. First, only one Brock-Mirman paper is ever cited as having been presented and discussed at the MSSB meeting, that is, their paper entitled "Optimal economic growth and uncertainty: the Ramsey-Weizacker Case", that is to say, the

"undiscounted case". This was later published by Brock and Mirman in the October 1973 issue of IER under the title "optimal economic growth and uncertainty: the no discounting case", according to Mirman (9

November 2014, personal communication to authors). Second, the

"discounted" case, that is to say the now classic 1972 Brock-Mirman JET paper was "received June 28, 1971" by the journal (1972, 479), or in other words, immediately prior to the July-August 1971 MSSB 40 conference. Finally, as will be seen below, the Brock-Mirman

"discounted" case paper was first circulated as a Rochester/Cornell

"mimeo" in 1970, and also presented at an NBER conference at Yale led by Stiglitz, and at the North American Meeting of the Econometric

Society in December 1970. And thus, it is to the intriguing story of the evolution and impact of the Brock-Mirman approach that we now turn.

3. Brock and Mirman: from thesis to meta-synthesis, 1970-1973

Brock and Mirman's 1972 JET and 1973 IER papers are well known, and the former is widely cited. What is less well known is that they also published another joint paper on the stochastic growth model. As

Mirman wrote (personal communication 13 June 2014) "there is a third

Brock-Mirman paper, which mirrors the discounted case paper [JET

1972]". This paper, albeit not widely cited, was presented at an October

1971 conference on "optimization techniques", and will be discussed below. All three Brock-Mirman papers were the outcome of Mirman's

1970 thesis and their collaboration in developing, as they put it "the unification" of previous approaches to growth (1972a, 483) or, in other words, their meta-synthesis. But before proceeding to a discussion of the evolution of the joint work, we deal with the development of Mirman's approach to uncertainty and growth, his 1970 thesis, and early papers.

Mirman, uncertainty, and growth: Chicago and Rochester 41

Mirman's interest in issues surrounding uncertainty and growth started as a graduate student. He recalled (personal communication,14

June 2015) that his interest in the study of uncertainty in economics

"…was a big problem for me early on, so many people -- including

Hirofumi Uzawa and David Cass -- disparaged my interest in uncertainty… ". Mirman recalled that in 1967 he went to the summer workshop on growth at University of Chicago organized by Uzawa, also attended by Ethier, Calvo, and Razin, among others. Mirman also wrote

(personal communication 15 June 2014):

I was a graduate student, at the end of my second year, just

finishing my course work, when I was sent by the department (or

by McKenzie) to a summer workshop on growth at the University

of Chicago run by Hiro Uzawa. In my early discussions with Hiro

about my work he was dead set against my working on uncertainty

and growth, I had some very preliminary ideas but needed much

more thought and work and help before anything could develop

from these ideas. I remember he thought that putting uncertainty in

a growth model was too hard and not enough was known about

certainty and growth to waste time on uncertainty. He didn't let me

work on uncertainty all summer, he had me working on a project

he found interesting but I had no idea what he was talking about. 42

At the end at the summer I was at wits end because I needed a

second year paper for Rochester. Luckily the only thing that made

sense to me then was uncertainty and I was able to write a paper on

an uncertain Von Neuman growth model. It mimicked the work

McKenzie was doing and it began to lay the foundation for my

thesis. In any case, I had met Cass at the conferences run by

Uzawa and discussed my ideas with him. In my view both Cass

(who was a student of Uzawa) and Uzawa were (are) brilliant

economists. In fact, Uzawa was a brilliant mentor, too bad he had

no use for uncertainty. But lucky for me that was my only idea.

Mirman's thesis and early papers

In their JET paper, "Optimal Growth and Uncertainty: the

Discounted Case", Brock and Mirman wrote (1972a, 482) "The basic framework for the paper was developed by Mirman in [9, 10], for discrete time one-sector stochastic growth models". Reference 9 was to Mirman's

1970 Ph.D thesis, entitled "Two Essays on Uncertainty and

Economics"(Mirman, 1970a) ; reference 10 was to Mirman's as yet

"unpublished" 1970 paper, entitled "The Steady State Behavior of a Class of One Sector Growth Models with Uncertain Technology"(1970b), which emanated from his thesis, and was later published in JET in 1973. 43

With regard to his thesis and influences on it, Mirman recalled

[personal communication 9 February 2007]:

Brock and I overlapped at Rochester, he as an assistant professor

and I as a graduate student, for one semester, and he told me he had

no idea what I was doing, but we were friends. I turned in my

thesis the next spring—I was at Cornell—and he was assigned to

read it. My advisor was McKenzie—Brock and Zable were on my

committee from economics—but the biggest help I got was from a

mathematician named Kemperman, who was also on my

committee. A statistician—who taught me stochastic processes--

was also on the committee, Keilson. I was really lucky to be

surrounded by a group of great scholars.

Another paper Mirman wrote at the time was his 1971 Econometrica paper entitled "Uncertainty and Optimal Consumption Decisions", received in March 1969; the final revision dated November 1969. In the first note to the paper, Mirman acknowledged "the encouragement and advice of Prof. J.H.B. Kemperman" (1971, 179).

In January 1971, Mirman sent his second thesis paper entitled "On the existence of steady state measures for one sector growth models with uncertain technology" for publication to IER, and after revision in

September 1971, it was published in June 1972. Now, in his IER paper, 44

Mirman cited his then "unpublished" 1970 thesis paper "The steady state behavior of a class of one-sector growth models with uncertain technology", as having introduced "a stochastic generalization of the concept of a steady state equilibrium for a model of economic growth"

(1972, 271). However, in his IER paper, Mirman did not cite his own

1970 thesis [as against its citation in Brock-Mirman (1972)] , and in the

IER paper, Mirman cited his JET paper with Brock as "forthcoming"

(1972, 286).

Mirman's 1970 paper, "The Steady State Behavior of a Class of One

Sector Growth Models with Uncertain Technology", finally appeared in the June 1973 issue of JET. When asked about the differential citations,

Mirman replied [personal communication 8 February 2007]: "This is easy to answer. I think my thesis paper and the…IER paper are almost exactly the same so there was no need to quote the thesis, but then I needed to quote the not yet forthcoming JET paper [our emphasis, as Mirman is talking here about the June 1973 JET version of his 1970 paper]. But the

1973 JET paper and the thesis paper are different—the referee insisted that I change the proof."

And indeed, in the introductory note to the June 1973 JET version of his 1970 paper, Mirman wrote: "This paper contains results reported in my Ph.D thesis…However, the organization of the paper and the proofs 45 of the main theorem have undergone considerable change". In the references to his 1973 JET paper, which as Mirman wrote (1973, 219), was based on his 1970 Ph.D thesis, both Cass (1965) and Koopmans

(1965) are cited, as is Radner (1971). Brock and Mirman (1972), however, only cite Cass and Koopmans, and a paper by Brock entitled

"Sensitivity of Optimal Growth Paths with Respect to a Change in Final

Stocks" [actually published (Brock, 1971) in the same conference volume as Radner (1971)].

Brock-Mirman papers: recollections, presentations, meta- synthesis

Over the period 1970-1973, the interaction between Brock and

Mirman took place on two levels. The first was the relationship emanating from Brock being Mirman's thesis examiner and intellectual catalyst for extending his approach. The second was as partners in presenting their joint papers at conferences, and publishing them so as to achieve the widest possible audience for what they described as "the unification" of approaches to growth their framework provided.

Above, we discussed the 1973 and 1975 versions of Merton's paper

"An asymptotic theory of growth under uncertainty". In both versions he cited a paper by Brock and Mirman entitled " The stochastic modified golden rule in a one-sector model of economic growth with uncertain 46 technology", as being published in the June 1972 issue of JET (1973, 33;

1975, 392). Moreover, Samuelson (1976, 491) also cited the 1972 JET

Brock-Mirman paper under the same title used by Merton, that is "The stochastic modified golden rule in a one-sector model of economic growth with uncertain technology". The title of the oft-cited 1972 Brock-

Mirman 1972 JET paper however was " Optimal economic growth and uncertainty: the discounted case".

In their citations, Merton and Samuelson actually referred to the title of the earlier versions of Brock and Mirman's watershed paper, which was initially circulated and presented at conferences in 1970. The paper, under its original title, first appeared as a "mimeo" emanating from

"Rochester and Cornell"; Mirman by then at Cornell, Brock at Rochester.

Recollections

In a series of queries from the authors, and replies from Brock and

Mirman, they recounted the evolution of their 1972 JET and 1973 IER papers, which they identified as "Brock-Mirman I" and "II" respectively

(1973, 560 footnote 2). When asked about the origins and dissemination of the 1970 "Cornell/Rochester mimeo"(1970a) and the first version of the 1972 JET paper, Mirman recalled (personal communications 12 June and 10 August 2014): " I do remember that we wrote, what is from hindsight, a preliminary version of the [1972 JET]… The first version of 47 the paper was sent out to a rather wide audience and we did get comments on it" (12 June). He went on to say (10 August)

I believe that my first communication with Brock …was in the

spring of 1970. I was to defend my thesis in May at Rochester and

he was a member of the committee and had just read the thesis. He

told me he had an idea for extending my thesis, which was for a

positive growth model under uncertainty, to the optimal case. My

work used the notion of stochastic technical progress, which was

taken from the work of Mirrlees that inspired me…to study the

question of growth under uncertainty…

Brock, for his part, recalled (personal communication 12 June 2014):

Rochester was my first job. I had just arrived there. I was reading

Len's thesis and I got very excited about the possibility of

generalizing his work on the stochastic Solow type model to the

infinite horizon optimization case. I recall not only talking to Len

by phone about this problem, but also going to Cornell to visit him,

where we worked on this problem and probably the undiscounted

one too…I recall it [the 1970 version of the 1972 JET paper] being

well known to researchers in growth before it actually was

published. 48

In a subsequent communication (13 June 2014) Mirman expanded his account and recalled:

I was an assistant professor at Cornell and just finished my

defense, when I received a call from Brock who was relatively new

in the department at Rochester. He told me that he had been

assigned to read my dissertation and was upset because he had just

finished reading a dissertation that was very boring and

uninteresting. He thought mine would be the same. He told me he

was very surprised that my dissertation was very interesting; in fact

he was sure that we can generalize it to the optimal case, and he

asked McKenzie why he had never told him about my work. My

dissertation studied the steady state behavior of a stochastic growth

model, similar to what Solow did in his classic paper for the

deterministic growth model. So Brock and I did the optimal case,

both in the discounted case and in the non-discounted case, the

latter never got the audience of the former.

When asked about the citations by Samuelson and Merton and impact of the published paper, Mirman replied (personal communication

14 June 2014): 49

I do remember that Brock got a short note from Samuelson. He

praised the work and introduced his student Merton, who was

working on similar problem in continuous time. Actually, over

time, it turns out that not many people actually read the published

version, for several reasons…

Mirman continued, relating the paper to his later work with Zilcha4:

There, I think, are two main reasons which are related. The first is

that the paper is well known and taught a lot so people think they

understand and know very well what's in it. The role of the other is

that the books of Sargent cover it in a way that people don't think

they need to read it. For example, our paper does not contain any

examples. But most people think that a Brock-Mirman model is

with log utility and Cobb Douglas production. This is from my

paper with Zilcha, and is presented by Sargent as the Brock-

Mirman model. So many people I've spoken with have never read

the original.

When asked about conference presentations of their 1972 JET paper,

Mirman recalled (personal communication 10 August 2014):

4 The Mirman-Zilcha growth model (1975) is based upon log utility and Cobb-Douglas production with exponential uncertainty. In three seminal papers, Mirman and Zilcha (1975; 1976; 1977) extended and amended the original Brock-Mirman model. Here, we only deal with the origins of Brock-Mirman, and thus we do not discuss the Mirman-Zilcha papers. However, the Mirman-Zilcha papers deserves careful reading by those applying the Brock-Mirman framework. 50

Brock came down to Cornell, where we wrote (what is now) a

barebones version of our paper. This is the paper, "The stochastic

modified golden rule in a one-sector model of economic growth

with uncertain technology". In the fall…we went to a conference

on growth theory at Yale at which Brock gave our paper, in a

pretty rough form.

Mirman continued:

In attendance was among others Karl Shell and David Cass. Karl

was at MIT the time and I believe that is how the paper got to

Samuelson (but I conjecture here)… In the meantime Brock and I

corresponded by mail and the paper started to expand with many of

the cracks and holes filled in. I would be remiss not to mention

that every time he sent me the paper I would send it back with the

key inequalities backwards, which drove him crazy… In the

meantime Brock and I worked on the paper.

Mirman went on to say:

My next recollection is a visit by Brock to Cornell, I was writing a

proposal for the NSF and he suggested that the introductory

material for the grant application be part of the paper. But at the

same time Brock had an idea for a second paper, the no discounted 51

case. We realized at this time that the work was more general than

just the stochastic technical progress case and that the dynamics

played an important part. So we decided that the reference to

modified golden rule was too narrow. We then changed the title to

take account of the generality of the paper and to link it to the

second paper. We continued to fill cracks and holes, as well as

added the appropriate diagrams. I think that it was in this form

with the name change that we sent it to JET.

Presentations

Below, we complement the recollections of Brock and Mirman by reference to the published record of conference presentations of the three

Brock-Mirman papers.

Brock-Mirman I

With regard to their 1972 JET paper, they recalled that it was first presented at Yale in in fall 1970 at a conference on "growth theory". An examination of the record of 1970 conferences shows that the paper was actually presented in November 1970 at the NBER conference on econometrics and mathematical economics funded by the NSF, and organized by Joseph Stiglitz (NBER Record, 1970). 52

The 1970 version of their 1972 JET paper was presented to a wider audience at the December 1970 North American regional conference of the Econometric Society held in Detroit (Econometrica, July 1971, 300) at session 15 "Growth Models", chaired by Stiglitz. The abstract of their paper "The stochastic modified golden rule in a one-sector model of economic growth with uncertain technology" (1970b) read as follows

(Econometrica, July 1971, 345-46):

In a discrete time one sector model of economic growth with

uncertain technology we show that the distribution functions of

optimal consumption and capital stock at time t (which are random

variables) converge pointwise to limit distributions. The limit

distribution, F, of optimal capital stock is a natural generalization

of the modified golden rule. Hence our result is an extension of the

Cass, "Optimal Growth in an Aggregative Model of Capital

Accumulation," RES 1965; Koopmans "On the Concept of Optimal

Economic Growth," Pontficae Academiae Scientiarum Scripta

Varia, 1965, results to the case where technology is uncertain. All

of our assumptions are similar to Cass-Koopmans except for the

random technology and the planning objective which is assumed to

be maximization of the capital value of the discounted sum of

utilities. We assume that technology can be represented by F(K, L, 53

r) where F satisfies the usual assumptions for each value of the

random variable r, and increases in r for each K, L. We have found

an elementary set of techniques to deal with this rather difficult

problem. First we follow Levhari and Srinivasan, RES 1969, and

use dynamic programming to establish the existence of time

invariant policy functions, giving optimal consumption and capital

stock at time t+1 as a function of capital in existence at time t.

Hence the evolution of optimal capital stock is given by a

stochastic process. This stochastic process is shown to converge in

distribution by exploitation of the necessary conditions of

optimality.

Brock-Mirman II

In his 1973 JET paper "Optimal stationary consumption with stochastic production and resources", Radner cited a paper by Brock and

Mirman entitled " Optimal economic growth and uncertainty: the

Ramsey-Weizacker case" as Working Paper number 7, emanating from the Mathematical Social Science Board Workshop on "The theory of markets and uncertainty", held at the Department of Economics,

University of California, Berkeley, in 1971. As noted above, this paper became their October 1973 IER Paper "Optimal economic growth and 54 uncertainty: the no discounting case" which was received at that journal in January 1972, and revised in November 1972 (1973, 560).

Brock-Mirman III

In October 1971, what Brock and Mirman have called "Brock-

Mirman III" was presented by Brock at the 4th International Federation for Information Processing colloquium on optimization techniques, Los

Angeles, 19 -22 October. This paper was entitled "A one-sector model of economic growth with uncertain technology: an example of steady state analysis in a stochastic optimal control problem". It was published in the

1972 conference volume techniques of optimization edited by

Balakrishnan (Brock and Mirman, 1972b, 407-19).

What is interesting about this sparsely cited paper, is that it is what could be considered an "executive summary" of their more extensive

1972 JET paper. This is evident in the introduction, literature survey, and concluding remarks of the paper. For example, in the introduction to

Brock-Mirman III, they pointed to their extension of Cass-Koopmans to the case of stochastic "output or technical progress", while departing from the Cass-Koopmans approach, as Brock and Mirman deal with "uncertain technology or technological progress", and this "in discrete time"; thereby establishing "a stochastic analog" of steady state convergence, which they termed "the modified golden rule". They then described their approach as 55

"essentially a blend of dynamic programming and discrete-time stochastic optimal control theory" (1972b, 407).

In their literature review, they cited the work of Mirman [1970, Ph.D thesis] and Mirrlees [1965] as "most relevant". However, they went on to say that "Mirrlees operates in continuous time where the theory of stochastic processes is messy", counterpointing this by then saying "we avoid the messy mathematics by working in discrete time" (1972b, 408).

In their concluding remarks they cited Mirman's 1971 University of

Massachusetts paper "The steady state behavior of a class of one-sector growth models with uncertain technology", which became his 1973 JET paper, and their own summer 1971 MSSB Working Paper 7, which became their 1973 IER Paper on the "non-discounted case". They also referred to Jeanjeans' 1971 Ph.D thesis as extending some of "their results to multisector models". Finally, they concluded by saying that in Brock-

Mirman III "they chose to take the self-contained route in order to delineate the ideas and to reach a wider audience" (1972 b, 418).

Metasynthesis

A close reading of Brock and Mirman's seminal 1972 JET paper reveals three major issues they dealt with: (i) dynamics of optimal processes and steady states; (ii) the use of dynamic programming; and 56

(iii) the synthesis of approaches to optimal growth. The first and second issues were set out by Mirman in a communication to the authors (12

August 2014). He wrote that in the 1972 Brock-Mirman paper there were:

Two issues that needed to be dealt with. The first is the dynamics

of the optimal process and its corresponding steady state. Mirrlees

[1965] dealt with an economy that had a concave technology and

thus, although not done, could deal with the stochastic dynamics

and corresponding steady state as, say, Cass [1965] does in the

deterministic case. The work of Levhari and Srinivasan [1969] and

Phelps [1962] deals with a linear technology. Hence, although it

might have, the issue of the dynamics and steady state does not

arise… The second issue is the use of dynamic programming.

Mirrlees does not use dynamic programming techniques. He takes

a deterministic "Euler conditions" and linearizes them… Both

Levhari-Srinivasan and Phelps use dynamic programming

techniques, in a very rudimentary form to get at the results, which

were done in a linear technology setting. Hence their method at

getting at the optimal program is foundationally similar to ours.

The third issue relates to Brock and Mirman's "unification", that is to say, "metasynthesis" of previous approaches to optimal growth. This is 57 expressed in what we take to be the central message of their 1972 JET paper, in a paragraph which, in our view, has been overlooked by most observers up to now, possibly reflecting the situation that they may not have actually read the full text of the paper, as Mirman noted in his recollections cited above. They wrote (1972a, 483):

The model used in this paper is analogous to the Mirrlees and

Mirman model of a one sector economy under uncertainty, which

is essentially the generalization of the Cass-Koopmans model with

a random variable in the production function. In fact, our methods

unify the structure of growth theory. The dynamic programming

formulation makes the Cass-Koopmans results somewhat easier to

obtain. It is thus seen that this paper represents a nontrivial

extension and unification of the work of Cass, Koopmans, Mirman

and Solow. [Our emphasis]

But let us leave the last word regarding the impact of their

"unification" to Lucas, who recognized the Brock-Mirman approach as one of the starting points for Kydland and Prescott's own "metasynthesis" that led to quantitative or empirical macroeconomics. Indeed, as Lucas put it "technically the immediate ancestor of Kydland and Prescott" was the Brock-Mirman 1972 JET paper (1987, 32 note 1). Just how Brock 58 and Mirman's approach influenced subsequent developments in growth and cycle theory is another story (see Young 2014, Chapter 1).

Conclusion

With the unification of growth theory around the three elements of optimization, dynamic programming and stochastic control, the modern development of the neoclassical growth model reaches completion in the work of Brock and Mirman. The model that now bears their name has become a workhorse in real business cycle theory and is the basis for models of repeated games and optimal taxation. The basic techniques of stochastic dynamic programming have spun off from the Brock-Mirman nexus into areas of applied microeconomics, finance, contract theory and even in dynamic game theory. But, the underlying theory encapsulated by the model has not changed in the forty plus years since the Brock-

Mirman paper appeared.

It is worth reflecting, then, on what the model and theory deliver, because it is only against the backdrop of these results that we can begin to understand why the neoclassical framework has fallen short as a theory not just of growth but also of micro-founded macroeconomics. The key results we associate with the neoclassical model are 59

• optimality – the equilibria generated by the model satisfy the first

welfare theorem, generating Pareto optimal outcomes;

• determinacy – the application of dynamic programming converts

the model into one amenable to analysis via concave programming,

so that optimal trajectories of the model have a saddle-path

property that implies the solutions are unique;

• ergodicity – under reasonable specifications of the discount factor,

the deterministic steady-state of the model is locally stable; hence

the stochastic extension of the model will exhibit ergodic behavior

asymptotically.

While these features seem eminently reasonable (and seem to reflect the fundamental results one obtains from the static Arrow-

Debreu model of general equilibrium), when we confront these results with empirical facts, the shortcomings of the model become apparent.

On the question of optimality, particularly in the context of real business cycle theory, the optimality of equilibria means that there can be no such thing as involuntary unemployment of input resources.

During the period of the 1980’s through the first decade of the 2000’s, the so-called “great moderation” in the world macroeconomy relegated this issue to a back burner, since those spells associated with downturns in the business cycle were short, and the world economy, 60 for the most part, spent most of the time near the full-employment threshold. It was also easy to ignore the experience of Japan in the

1990’s, blaming their woes on demographic effects or the peculiarities of government regulation. But the financial crisis of 2008 and the subsequent long-lived downturns in the U.S. and Europe saw the economics profession arguing with itself (as it had in the 1930’s) over whether or not there was anything to be done about the slump.

Proponents of real business cycle theory stated clearly that the observed unemployment resulting from the crash was entirely due to a marked reduction in productivity that made continuing to work undesirable. This, of course, is part and parcel of the so-called freshwater-saltwater divide in macroeconomics, but the distinctly unmoderated effects of the ’08 crash and its aftermath have brought the optimality implication of the RBC model and its neoclassical underpinnings to the fore.

A second issue that RBC macro has pushed to center stage is the question of what gives rise to aggregate shocks. While the Brock-

Mirman assumption was, as a theoretical construct, entirely acceptable, confronting the mechanism with actual data in the calibrated versions of Brock-Mirmin pioneered by Kydland and

Prescott has posed problems. Specifically, actual shocks large enough 61 to impact the economy as whole (particularly large economies such as the U.S. or EU) don’t occur at anything close to business-cycle frequencies. Sectoral shocks do occur more frequently, but the connectedness of input-output relationships between different sectors leads to the conclusion that the law of large numbers should dampen the overall effect of these shocks on the economy as a whole. So, despite the success of RBC models in explaining many macroeconomic co-movements, the question remains as to what actually drives business cycles.

Finally, as a model of economic growth (independently of any other applications of the model or its methodology), the neoclassical growth model never actually moves beyond Solow’s original work and its finding that, except for population growth, nothing in the neoclassical model explains the economic growth experienced since the onset of industrialization on the mid-1700’s (i.e. what we now routinely refer to as the Solow residual). This problem is particularly galling, since it means that all of the work that went into the development of optimal growth theory can’t actually explain growth, optimal or not. And, in this problem, we find the seeds for the development of a new theory of growth which ultimately leads to the unraveling of the key features of the neoclassical growth model, as 62

optimality gives way to equilibrium in environments in which the

perfect competition and complete markets assumptions of the static

Arrow-Debreu model, and it’s dynamic extension in the neoclassical

model must give way to increasing returns and knowledge

externalities. We pursue this topic in our next paper on the

endogenous growth revolution.

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